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Cayley diagram

Q8×C22⋊C4 #2340

order128
nilpotency class2
isoclinismisoclinic class of D42
automorphismAut(G) order 49152
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productQ8×C22⋊C4

Subgroups

subgroupG/H
center, Frattini, socle, cosocleC23C24
commutatorC22C23×C4
prime radicalC24C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

(9,10,12,13)(11,15,16,14);(9,11,12,16)(10,14,13,15);(2,4)(5,6,7,8)(9,13,12,10)(11,15,16,14);(1,2)(3,4)(5,6)(7,8)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC22⋊C4×SL(2,𝔽3), #11711198, #11711199, …
maximal subgroup of#9258, #9262, #22437, …
maximal normal subgroup of#9258, #9262, #22437, …
maximal quotient ofC22⋊C4×C4⋊C4, Q8×#95, #13817, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1072);